\documentclass[12pt]{article} \usepackage{amsmath,amsthm,amsfonts,amssymb,latexsym, hyperref} \begin{document} \begin{center} \textbf{Groups} \end{center} Definition (reminder) If $n \in \mathbb{N}$ then for $a, b \in \mathbb{Z}$ we define the equivalence relation $\equiv_n$ on $\mathbb{Z}$ as follows: $a \equiv_n b$ if and only if $n | (b-a)$; $\mathbb{Z}_n$ denotes the equivalence classes: $\mathbb{Z}_n = \{[x]_n | x \in \mathbb{Z} \}$. \ Theorem 9.0. Define the operation $+_n$ and $\cdot_n$ on $\mathbb{Z}_n$ as follows: $$[x]_n +_n [y]_n = [x+y]_n$$ $$[x]_n \cdot_n [y]_n = [x \cdot y]_n$$ Then the operations $+_n$ and $\cdot_n$ are well defined. \ Exercise. Consider the objects $\mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_4, \mathbb{Z}_5, \mathbb{Z}_6, \mathbb{Z}_7$ with the operations $+_n$ and $\cdot_n$. Construct the addition and multiplication ``tables''. We will be making heavy use of these objects. \ A group is a set of elements $G$ with an operation $\cdot$ that has the following properties: 1. Closure: if $x \in G$ and $y \in G$ then \qquad \qquad $x \cdot y \in G$; 2. Associativity: if $x, y, z \in G$ then \qquad \qquad $(x\cdot y) \cdot z = x \cdot (y \cdot z)$; 3. Identity: there is an element $e \in G$ so that for each $x \in G$: \qquad \qquad $e \cdot x = x = x \cdot e$; 4. Inverses: for each $x \in G$ there is an element $x^{-1}$ so that \qquad \qquad $x \cdot x^{-1} = e = x^{-1} \cdot x$. \noindent Caution!: Group operations are not necessarily commutative. Example: the set of all $n \times n$ matrices with non-zero determinants form a non-commutative group under matrix multiplication. \ Theorem 9.1 [Uniqueness of the identity]. Suppose that $G$ is a group with identity $e$. If $\hat e$ is an element of $G$ so that for all $x \in G$, $ \hat e x = x = x\hat e$ then $e = \hat e$. \ Theorem 9.2 [Uniqueness of the inverse]. Suppose that $G$ is a group with identity $e$ and $x \in G$. Then there is a unique element $x' \in G$ so that $x \cdot x' = x' \cdot x = e$. [Notation: the unique inverse of the element $x$ is denoted by $x^{-1}$.] \ Theorem 9.3. Suppose that $G$ is a group and $x, y, z \in G$ are arbitrary elements. Then: \qquad 1. $(x^{-1})^{-1} = x$. \qquad 2. $(xy)^{-1} = y^{-1} x^{-1}$. \qquad 3. $(xy = xz) \Rightarrow (y = z)$. \qquad 4. $(yx = zx) \Rightarrow (y = z)$. \ Example 9.1. Let $S = \{1, 2, 3, \dots, n\}$ define $S_n$ to be the collection of all 1-to-1 functions of $S$ onto itself. Define the operation $\circ$ between the elements $\alpha, \beta \in S_n$ by ordinary composition, thus for each $s \in S$ we have $(\alpha \circ \beta)(x) = \alpha(\beta(x))$. The set $S_n$ with the operation $\circ$ is a group. \ Definition. A group $G$ is said to be Abelian (or to be a commutative group) if and only if $xy = yx$ for all $x, y \in G$. \ Exercise 9.1. Construct the multiplication charts for the groups $S_2$ and $S_3$. Are these groups Abelian? \ Exercise 9.2. How many elements are in the groups $S_4$ and $S_5$. Show that these groups are not Abelian and that each one of these has a ``subgroup'' equivalent to $S_3$. \ Definition. Suppose that $G$ is a group with operation $\cdot$ and $H \subset G$. Then $H$ is said to be a subgroup of $G$ if $H$ with the operation $\cdot$ is a group. \ Exercise 9.3. We would like to determine when the following sets with the indicated operations are groups, assume $n$ is an integer with $n>1$: \begin{eqnarray*} \mathbb{Z}_n & \mbox{with operation} & + \mod n \\ \mathbb{Z}_n - \{[0]\} & \mbox{with operation} & \cdot \mod n. \end{eqnarray*} Look at examples for $n=6,7,10,11$. Which of these yield groups (it's not necessary to write out the whole table to answer this question.) (And why was $0$ removed from the set?) \ Theorem 9.4. Suppose that $G$ is a group with operation $\cdot$ and $H \subset G$ and it it true that for $h_1, h_2 \in H$ we have $h_1 \cdot h_2^{-1} \in H$. Then $H$ is a subgroup of $G$. \ Notational conventions. When working with the set $\mathbb{Z}_n$, then I will frequently omit the brackets $[x]_n$ when it is understood that we are working with $\mathbb{Z}_n$; and operations $+_n$ and $\cdot_n$ are often denoted by $+ \mbox{ mod } n$ or $\cdot \mbox{ mod } n$ respectively. Thus following are equivalent ways of writing the same thing: \begin{eqnarray*} x \equiv_n y & \Leftrightarrow & x = y \mbox{ mod } n; \\ {[3]}_5 +_5 {[4]}_5 = {[2]}_5 & \Leftrightarrow & 3 + 4 = 2 \mbox{ mod } 5. \end{eqnarray*} \ Exercise 9.4. Find all the subgroups of $(\mathbb{Z}_6, + \mod 6)$ and of $(\mathbb{Z}_7 - \{[0]\}, \times \mod 7)$. \ Definition. For $\mathbb{Z}_n$ I want to be able to define the quantity $[b] = [a]^{[x]}$. Unlike the addition and multiplication operators this is not naturally well-defined. (In fact, as related in class, if $x<0$ then $a^x$ is not even an integer.) So we define it as follows: if whenever $x, y > 0$ we have that $x \equiv_n y \Rightarrow a^x \equiv a^y$ then we define $[b]_n = [a]_n^{[x]_n} = [a^x]_n$. When it is defined, we can let $[a^x]$ denote $[b]$ for positive integers $x$. \ Definition. Suppose that each of $G$ and $H$ are groups with operations $\otimes$ and $\boxtimes$ respectively and that $\varphi: G \rightarrow H$ is a function. Then $\varphi$ is called a \textit{homomorphism} if the following holds for all $x, y \in G$: \begin{eqnarray*} \varphi(x \otimes y) & = & \varphi(x) \boxtimes \varphi(y). \end{eqnarray*} A homomorphism that is 1-to-1 is called an \textit{isomorphism}. \ Exercise 9.5 Determine which of the following functions are well-defined, if so are they homomorphisms: (Note that I am abbreviating the elements of the groups so that, for example in a: $x$ means $[x]_6$, $\varphi(x)$ means $[\varphi(x)]_{12}$.) Are they isomorphisms? $$\begin{array}{rlll} a. \ \ \ & \varphi (\mathbb{Z}_6, +_6) \rightarrow (\mathbb{Z}_{12}, +_{12}) & \mbox{with} & \varphi(x) = 2x \mbox{ mod } 12\\ b. \ \ \ & \varphi (\mathbb{Z}_6, +_6) \rightarrow (\mathbb{Z}_{10}, +_{10}) & \mbox{with} & \varphi(x) = 2x \mbox{ mod } 10\\ c. \ \ \ & \varphi (\mathbb{Z}_{10}, +_{10}) \rightarrow (\mathbb{Z}_{5}, +_{5}) & \mbox{with} & \varphi(x) = 4x + 3 \mbox{ mod } 5 \\ d. \ \ \ & \varphi (\mathbb{Z}_6, +_6) \rightarrow (\mathbb{Z}_7 - \{0\}, \cdot_7) & \mbox{with} & \varphi(x) = 3^x \mbox{ mod } 7\\ e. \ \ \ & \varphi (\mathbb{Z}_6, +_6) \rightarrow (\mathbb{Z}_7 - \{0\}, \cdot_7) & \mbox{with} & \varphi(x) = 2^x \mbox{ mod } 7\\ f. \ \ \ & \varphi (\mathbb{Z}_6, +_6) \rightarrow (\mathbb{Z}_7 - \{0\}, \cdot_7) & \mbox{with} & \varphi(x) = 5^x \mbox{ mod } 7\\ g. \ \ \ & \varphi (\mathbb{Z}_{12}, +_{12}) \rightarrow (\mathbb{Z}_6, +_6) & \mbox{with} & \varphi(x) = x \mbox{ mod } 6 \end{array}$$ \ Notation. If $G$ is a group with identity element $e$ and $g \in G$ then: \qquad i. $g^0$ denotes $e$; \qquad ii. $g^1$ denotes $g$; \qquad iii. for a positive integer $n>1$, $g^n$ is defined inductively as: $$g^n = g^{n-1}\cdot g.$$ \ Theorem 9.5. Suppose that $G$ is a group with the usual notation for the operation. Then: $$\begin{array}{rlll} \mbox{a.} \ \ \ & (g^{-1})^n = (g^{n})^{-1}& \mbox{for } g \in G, n \in \mathbb{Z}^+ \\ \mbox{b.} \ \ \ & g^n \cdot g^m = g^{n+m}& \mbox{for } g \in G, n,m \in \mathbb{Z}^+ \end{array}$$ Observe that condition (a.) allows us to define $g^{-n}$ as the inverse of $g^n$. \ Exercise 9.6. Prove that if $G$ is a group and $g\in G$ then $H = \{ g^n | n \in \mathbb{Z} \}$ is a subgroup of $G$. Note: $H$ is called a \textit{cyclic subgroup} of $G$; if there is an element of $g \in G$ so that the corresponding subgroup $H$ is all of $G$ then $G$ is called a \textit{cyclic} group. \ Theorem 9.6. Suppose that $G$ is a group and $H$ is a subgroup of $G$. Define the relation $\sim$ on $G$ by $g \sim h$ if and only if $g h^{-1} \in H$. Then: \qquad a. $\sim$ is an equivalence relation on $G$. \qquad b. Let $p \in G$ and define $Hp = \{ hp | h \in H \}$; then the function $f: H \rightarrow Hp$ defined by $f(h) = hp$ is 1-to-1 and onto. Definition: the set $Hp$ is a called the \textit{right coset} of $H$ generated by $p$. \qquad c. $[e]_{\sim} = H$. \qquad d. The collection $\{ Hg | g \in G \}$ is a partition of $G$. \ Exercise 9.7. Consider $G = (\mathbb{Z}_{12}, +)$. Let $H = \{ 0, 3, 6, 9 \}$. \qquad a. Show that $H$ is a subgroup of $G$. \qquad b. Find all the cosets of $H$ in $G$ and denote this set by $G/H$. [Note: If $x \in G$ then $H +_{12} [x]_{12} = \{[h + x]_{12} \big| [h]_{12} \in H \}$ is the coset generated by $x$.] \qquad c. For $H +_{12} [x]_{12}, H +_{12} [y]_{12} \in G/H$ define $(H +_{12} [x]_{12}) \oplus (H +_{12} [y]_{12})$ by $(H +_{12} [x]_{12}) \oplus (H +_{12} [y]_{12}) = H+_{12}[x+y]_{12}$. \qquad d. Show that $\oplus$ is well defined and construct the addition table for $G/H$ with the operation $\oplus$. \noindent Let $\varphi: G \rightarrow G/H$ be defined by $\varphi(x) = H +_{12} [x]_{12}$. \qquad e. Is $\varphi$ well defined? \qquad f. Is $\varphi$ 1-1 and/or onto? \qquad g. Is $\varphi$ a homomorphism? - an isomorphism? \ Corollary to 9.6 [Lagrange's theorem]. If $G$ is a group and $H$ is a subgroup of $G$ then $|H| \Big{|} |G|$. \ Theorem 9.7. Suppose that $G$ is a group and $g \in G$. Then the set $H = \{h | gh = hg \}$ is a subgroup of $G$. \ Theorem 9.8. Suppose that $G$ is a group. Let $H = \{h \in G | gh = hg \mbox{ for all } g \in G \}$ is a subgroup of $G$. (This is called the commutator subgroup of $G$ and is the set all elements that commute with all the elements of $G$.) \ Theorem 9.10. Suppose that $G_1$ and $G_2$ are groups and $\varphi: G_1 \rightarrow G_2$ is a homomorphism. Then $h(e_1) = e_2$ where $e_1$ is the identity element of $G_1$ and $e_2$ is the identity element of $G_2$. \ Exercise 9.8. Consider the group $(\mathbb{Z}_n, +_n)$ with the operation of addition $\mod n$. Suppose that $H$ is a subgroup of $\mathbb{Z}_n$. Let $J$ be the collection of all cosets of $H$ in $\mathbb{Z}_n$. Define the operation $\oplus$ on $J$ as follows: $$(H +_n x) \oplus (H +_n y) = H +_n (x+_n y).$$ Define the operation $\boxplus$ as follows: if $H_1$ and $H_2$ are two cosets then $$ H_1 \boxplus H_2 = \{ x +_n y | x \in H_1, y \in H_2 \}.$$ Show that: a. $\oplus$ is well defined. b. $H_1 \boxplus H_2 = H_1 \oplus H_2$. b. $J$ with the operation $\oplus$ is a group. c. $J$ is abelian. d. $|H| \cdot |J| = n$. \ Exercise 9.9. Prove that a group $G$ is abelian if and only if $(xy)^2 = x^2 y^2$ for all $x,y \in G$. \end{document}